8.4: Elliptic, Parabolic, and Hyperbolic Points

Defined in terms of the Second Fundamental Form

A direction in the tangent plane at a point for which the normal curvature is zero is called an asymptotic direction. In terms of the second fundamental form, such a direction satisfies the condition

L11 + 2L12λ + L22λ2 = 0

where λ = v'(t)/u'(t). If we have a curve on the surface defined by X(t), with a non-zero tangent vector, this condition can be written as II(X'(t),X'(t)) = 0. By the quadratic formula,

λ = [-2L12 ± (4(L12)2 - 4L11L22)]/(2L22)

λ = (-L12 ± -det(Lij))/L22

If at some point P of a given surface det(Lij) > 0 we call P an elliptic point. Clearly at such a point P there can exist no real asymptotic directions. If det(Lij) = 0 at P , P is called a parabolic point of the surface. Parabolic points can only have one asymtotic direction. For example, all points on a cylinder are parabolic and only have one asymptotic direction, namely the straight line. If det(Lij) <0 at P , P is called a hyperbolic point and P will have two asymptotic directions.


Defined in terms of Gaussian Curvature

While many differential geometry books use the definitions provided above for elliptic, parabolic and hyperbolic points of a surface in three-space, others define elliptic, parabolic and hyperbolic points of a surface to be points where the Gaussian curvature is strictly positive, zero and strictly negative, respectively. The connection between the two definitions can be explained using the Weingarten map. The coefficients of the second fundamental form are related to those of the Weingarten map by the relation (summing over repeated indices)

Lij=Likgkj

or, equivalently

Lik = Lijgjk

where gijgjk = δik , the Kronecker delta (i.e. gij is the inverse matrix of the first fundamental form). Since the det(Lik) = K(u,v), and since we have seen that det(gij)≥0, the Gaussian curvature has identically the same sign as det(Lij).

Asymptotic Directions Demonstration

Depending on whether we take the plus sign or the minus sign in the above equation for the asympotic directions, we can have different sets of vectors. In this demo, we can show one set, or the other, or both. These display options are controlled by the Show Vec Field 1 and Show Vec Field 2 checkboxes respectively. We can move around a point in the domain, and adjust the tangent directions of two curves on the surface. The curves are shown on the surface, along with each of their normal curvature vectors κNN. Line up the tangent directions with the vector field to see that the normal curvature is zero in the asymptotic directions. (NOTE: does not work for horizontal and vertical directions).

Exercises

    1. The default option here is the graph of a quadratic polynomial. Look at the asymptotic directions on a torus, and the function graph of the perturbed monkey saddle.

    2. On the torus, what happens to the asymptotic curves between regions of positive and negative Gaussian curvature? Why?

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